Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus . . .

. . . how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a “group of transformations” or a “complex number”? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.

Students could learn math in the context of real-world problems, they writes.

Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.

Applied math students would learn abstract reasoning skills as well as useable knowledge, they argue. Teaching only abstract math is like teaching Latin instead of Spanish and German.

In math, what we need is “quantitative literacy,” the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and “mathematical modeling,” the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car).

Tracking is taboo in schools. I can envision massive resistance to splitting students into abstract math and applied math streams. And Common Core Standards enshrine the traditional math sequence as the way to teach all students.

Comments

This idea deserves a pilot test. You could enroll a bunch of kids whose Math achievement scores indicate they’ll end up in remedial math in college anyway, and offer them this “math literacy” (please don’t call it Math lite, because part of the deal would be rigor, I hope). Then at HS graduation, compare their scores with similar kids who had continued to slog thru Alg 1, Geometry, and Alg 2 (most of those kids don’t make it into Trig or Calculus anyway).

And by the way, this might have been a really good option for one of my kids, who sleep-walked her way through the Alg 1, Geometry, Alg 2 sequence and learned nothing at all, or nothing that she retained.

People who understand math well can actually learn all the formalism of algebra, calculus, etc… fairly quickly. It used to be all taught in a couple of years to students who were really good at math, but we opened it up to everyone.

The formalism of math is a barrier for students because, just like when people struggle with a language, they get stuck at the syntax, and don’t get to dive into actual mathematical reasoning.

I’d argue that everyone should do these courses and that a few students should learn the formalism based on interest.

The fact that a couple of testing companies are making money off of the current system shouldn’t stop us from doing what is best for kids.

The answer (as usual) is “it depends.” There’s nothing wrong with the idea of “applied math” to contextualize what students learn. But what the authors are proposing goes way beyond that. “Replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering” means, as I read it, project based learning, which was a revolutionary idea whose provenance dates back 100 years, and which keeps coming back over and over and over.

I remember a LOT of applied math in Precalc and Calc. We had tons of compound interest problems, related rates problems, finding the volume of every day shapes problems, physics issues, industrial planning problems…….

In fact, IIRC, the “real world applications” of what we’d learned were often MORE difficult than the other problems. Same for statistics, trig, etc.

So here’s my concern– if you really want to do applied math with high school kids, either you’re going to find that the struggling kids do even WORSE with the applied stuff (because it’s more irregular and takes MORE analytical skills, which are what the “bad at math” kids are often lacking anyway

OR

you’re going to produce contrived “real world applications” that are actually dumbed down and give kids the IMPRESSION that they can apply Math when really the methods only work in carefully controlled situations.

I’m currently teaching quantitative literacy at the community college level. The vast majority of students who take it need either this class or a basic stats class to fulfill a graduation requirement.

I have taught this class for several semesters over the past few years. My impression: Most students don’t really care for the class, despite its “relevancy.” It seems that not a few felt misled by their advisers, thinking that they were enrolling in an “easier” math class. They tend to wish that they had enrolled in the basic stats instead, because it is more straightforward as a math class. Of course, it doesn’t help that a number of them have put the class off as long as possible, and subsequently find themselves in the position that they have to pass the class NOW in order to graduate this semester. And (of course) since they feel under the gun, I wind up being the mean old teacher who is the only thing standing between them and graduation.

I really don’t see the advantages of having “quantitative literacy” instead of math. We’re still dealing with the same bunch of horses that simply don’t want to drink.

Oh, on other thing– it’s true that “applied Math” is easier than proofs ala Spivak —but most high school math classes already do hardly any proofs! Heck, just trying to find an “all proofs” geometry text is a pain, and then everyone warns that it’s only for “very advanced” students.

I think one discussion we ought to be having is “Why do we teach math at all?”

Is it for computation? To teach problem solving and analytical skills? To get kids to stretch and think? Because it’s an important signaling mechanism to colleges and employers? All of these, in some proportion or another?

Then, once we figure out what purpose math in the schools is mean to serve, we can see if there are other fields of study that would serve the same purpose.

Personally, I like math. Barring major cognitive disabilities, my husband and I expect all our kids to make it through AP Calc before graduating High School. But it seems like a lot of reformers can’t decide what they want math to be DOING. And until they narrow that down, all these random ideas for “Math reform” seem like exercises in “Throw everything against the wall and see what sticks.”

Except the second premise is crazy false. Except for Driver’s Ed and maybe basic 6th grade literacy, there’s nothing that schools teach that is regularly used by more than 50% of the population. Most adults never touch chemistry or physics. Most never use history for anything. “Most” adults probably don’t even ever write anything more than a paragraph in the course of their typical duties.

“We shouldn’t teach what most adults don’t use except to students who demonstrate readiness and willingness to learn it.”

I’d go for forcing the unwilling to take it if they maintain progress despite their unwillingness. Not all that many HS students crave to take more math, but many see a benefit to it. But for the ones that get through Algebra 1 with a D, despite effort, I’d say something else is in order.

With high school students, we’re dealing with an in-between age where we have to begin to let them follow their own interests and talents. In K-8, we spare no effort to get all kids thru the standard K-8 curriculum, even if some need a lot of extra help to get there. Once they’re in college or in the work force, they get to pursue what they want. In high school, I believe, we refuse to realize that many students are, in their heads, figuring out what they are good at and what they prefer to do. We may not always like their choices, but school doesn’t go well for many if we continue to cram them into the college prep box. Or, for that matter, in any box.

Many years ago, when business degrees were actually harder to obtain, courses like business calculus showed business majors how to use calculus in solving business problems.

Of course, it still required students to have a working knowledge of pre-calculus before they could take this course, but I’d say the hardest course math wise required of any student in a business major these days is perhaps economics statistics or applied stats (this course is usually an upper division course), but in reality usually covers math which is no harder than what should have been covered in any high school courses 25+ years ago.

Many studies have shown why students do not understand math, and in many cases, it starts right in elementary school (grades 1 through 5) where students are taught math not by subject matter experts, but many times by persons who do not understand math themselves.

I’ve been thinking about this today– we have kids learn math (even if they won’t use it) because math is cumlative and takes a long time to master. So even though very few adults will use Trig in their day to day life, you can’t go a lifetime without math, decide “I need trig” and work your way up to it quickly.

Math gives people options. The more you have, the more options you have. As a parent, I want all my kids to go through Calc at least, so that they will have the freedom to choose whatever career path interests them.

For the kids who never “get” algebra, there are automatically fewer options. So the question is, at what point to we cut their losses and say “Look kid, you’re never going to be a doctor or an architect or an engineer. Time to reevaluate.”

The current regime allows kids who can barely handle fractions to believe they can be doctors and then fail out of community college.

On the other hand, I don’t know how we’d be able to offer a “math for the hopeless” track without being accused of enforcing stereotypes….

… students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now…

Do they need to know what constitutes a “group of transformations” or a “complex number”?

If I recall my physics correctly, sound waves lead to Doppler shift and you could model it as a set of Galilean coordinate transformations; so you’d need it THERE at least, but there’s a nice context. TV signals and the workings of computers? Man, I remember all kinds of complex numbers there and plenty of 2nd through 4th order algebraic expressions. I mean, answering why the sky is blue requires a 4th order expression because of dipole radiation and diffraction or some such.

MAYBE what would work better would be math teachers with more training in applied science so they could draw upon this in the classroom.

Deidre: I agree with your last post. Regarding the last comment, if we dropped the mandatory schooling age to 14, we’d be dealing with fewer of the hopeless, since it’s likely they are unmotivated as well as mathematically clueless. We need to accept the fact that not all horses are willing to drink.

As a college professor, I’d be happy if someone would just teach them basic computational skills–addition, subtraction, multiplication, division, fractions, decimals, and percents–that they can do without a calculator by the time they graduate from high school. The average high school graduate does not need high school algebra, geometry, or any other advanced math course. Of course, we’d like them to have these courses. But, the average student does not need the courses nor will he master what is taught in the courses.

I’ve learned that kids don’t really *want* relevancy, they want “easy”. So-called word problems can be made very relevant, but so many students sure don’t like them!

On the other hand, why does everything *have* to be relevant? Bertrand Russell was quoted as saying, “The first task of education is to destroy the tyranny of the local and immediate over the child’s imagination.” Just a thought.

There’s no reason *not* to offer these suggested courses, but to think poor math students are going to be able to grok compound interest when they can’t understand the slope of a line seems like wishful thinking to me.

Anon…when I attended high school 30 years ago, most of the students knew their basic math facts w/out using a calculator (of course, affordable scientific calculators were just becoming available in 1979/80).

That’s exactly right, Bill. And in today’s “complex world” when we all need to have “21st century skills” (ain’t education jargon great?) that’s all the average person needs to be able to do–basic computation. The nonsensical notion that everyone needs algebra, geometry, trig, and calculus is a waste of time and money. Just one more of education’s “romantic” ideas, to reference Chas. Murray.

Do you think Messrs. Garfunkel and Mumford believe their children and grandchildren might have the smarts to become “professional mathematicians, physicists and engineers?” Why? Do they think the children of bus drivers have the smarts? Or should the children of bus drivers be glad to receive an education befitting “citizens?”

I’m not arguing against tracking in high school. I’m very leery of setting up systems which perpetuate class differences. How would the division between abstract-math-worthy and citizen be determined? If there’s a placement test, all you’ll do is unleash a frenzy of at-home test cramming amongst the affluent and educated. If it’s by teacher opinion, well, you won’t necessarily end up with the future high performers in the class. You will, however, end up with all the PTA officers’ children in the class.

Garfunkel and Mumford don’t acknowledge this, but we’ve been here before: back in 1920, when the Committee on the Problem of Mathematics, headed by William Heard Kilpatrick, argued that algebra and geometry should be eliminated from most courses of study. As Diane Ravitch describes it in “Left Back”:

‘The Kilpatrick committee recommended that mathematics be tailored for four different groups: first, the “general readers,” who needed only ordinary arithmetic in their everyday lives; second, students preparing for certain trades (e.g., plumbers or machinists), who needed a modest amount of mathematics, but certainly not algebra and geometry; third, the few students who wanted to become engineers who needed certain mathematical skills and knowledge for their jobs; and last, the “group of specializers,” including students “who ‘like’ mathematics,” for whom the existing program seemed about right, although the committee proposed “even for this group a far-reaching reorganization of practically all of secondary mathematics.”‘

Diedre and Katherine said all that I wanted to say, except for my usual:…
Federalism and markets institutionaliize humility on the part of State actors. If we disagree about a matter of taste, numerous local policy regimes and competitive markets in goods and services allow for the expression of varied tastes while the contest for control over a State-monopoly provider must inevitably create unhappy losers (who may comprise the majority; imagine the outcome of a nationwide vote on the one size of shoes we all must wear). If we disagree about a matter of fact, where “What works?” is an empirical question, a federal system, with numerous local policy regimes or a competitive market in goods and services will generate more information than will a State-monopoly provider. A State-monopoly enterprise is an experiment with one treatment and no controls, a retarded experimental design.

In __The Cancer Ward__ Aleksandr Solzhenitsyn meditates on the question, “when may one person prescribe for another?” and concludes “when there is a bonnd of love between them”. And it must be personal, and not some abstract “love of the People”. This applies to curriculum as well as medicine.

Within broad limits, young children should study what their parents want them to study, and as they get older, what they themselves want to study. Enough of us like Math, and Math is sufficiently rewarding, that Math-oriented parents will reproduce in numbers sufficient to systain a modern economy. Or not, if that’s what people want.